File::Copy can rename directories, but it can't copy them, so it can't be used to move one to another filesystem.

It's probably simpler to use system to execute a tool provided by your OS.

By the way, no need to put your variables in double quotes.

Update: How about:

my $cp_rv = system('cp', '-rp', $src, $dst);

if ($cp_rv) {
   if ($cp_rv == -1) {
      die("Unable to launch 'cp': $!\n");
   } else {
      system('rm', '-rf', $dst);
      die("Unable to copy \"$src\" directory. \"cp\" returned $?\n");
   }
}

system('rm', '-rf', $src);
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